Research Article

Advances in Mechanical Engineering2017, Vol. 9(8) 1–9� The Author(s) 2017DOI: 10.1177/1687814017717182journals.sagepub.com/home/ade

Casing failure mechanism duringvolume fracturing: A case study ofshale gas well

Tiejun Lin1, Hao Yu1, Zhanghua Lian1 and Biao Sun2

AbstractA large number of casing failures occur during the volume fracturing operation of shale gas, making normal completionstimulations impossible. To solve this problem, rock mechanical experiments and numerical simulation experiments arecarried out in this article. It is found that the macroscopic rock mechanical strength reduces most when the crack angleof fissured rock in Longmaxi Formation is 45�, and it reduces stably when the number of cracks increases to 8. The elas-ticity modulus ratio, yield strength ratio, and compressive strength ratio are 0.70, 0.71, and 0.68, respectively, based onwhich this article establishes the finite element model for shale gas well X201. Then, the secondary development realizesthe dynamic adjustment of the rock mechanical properties during the fracturing. The correctness of method and modelin the article is verified through comparing the simulated calculation of casing deformation and the field multi-arm caliperlogging data. The casing failure mechanism is revealed, providing a theoretical basis for the prevention of casing failurecaused by shale gas fracturing.

KeywordsShale gas fracturing, casing failure, finite element, numerical experiment, secondary development, multi-arm caliperlogging

Date received: 30 April 2016; accepted: 2 June 2017

Academic Editor: Farzad Ebrahimi

Introduction

Volume fracturing is a new technology that can ‘‘shat-ter’’ the reservoir, form complex crack networks, andcreate ‘‘artificial’’ permeability. Crack initiation in thetechnology is realized by shearing, breaking, and slip-ping. This technology breaks through the traditionalmode of crack percolation theory and significantlyshortens fluid flow distance. It has stimulated unconven-tional oil and gas production greatly and been widelyapplied to the transformation of rock layer with higherbrittleness. Meanwhile, it adopts the staged multi-clusterperforation.1 The proposal of ‘‘volume transformationtechnology’’ subverts the classic fracturing theory.2

Compared to conventional fracturing, volume frac-turing is characterized by excessive stimulated stages,large fracturing volume, and high injection capacity in

the operation process. The technique breaks up reser-voirs into pieces and forms complicated fracture net-works, which degrades the mechanical properties of

1State Key Laboratory of Oil and Gas Reservoir Geology and

Exploitation, Southwest Petroleum University, Chengdu, China2Engineering Technology Research Institute, Xinjiang Oil Field Company,

Karamay, China

Corresponding authors:

Hao Yu, State Key Laboratory of Oil and Gas Reservoir Geology and

Exploitation, Southwest Petroleum University, 610500 Chengdu, China.

Email: yuhao881126drill@163.com

Zhanghua Lian, State Key Laboratory of Oil and Gas Reservoir Geology

and Exploitation, Southwest Petroleum University, 610500 Chengdu,

China.

Email: milsu1964@163.com

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License

(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

formation rock much larger than the convention frac-turing. There will exist complicated mechanical beha-viors such as shear, leap, and slip around the casingstring, which makes casing more prone to failure. Inthe development of China shale gas, frequent collapsesand deformations of casing prevent the bridge plugfrom setting in the very place. The effect of fracturingoperation is mitigated as a result, and the running ofsubsequent tools gets harder. Along with it, construc-tion costs are increased and the construction becomestougher, making it difficult to ensure the wellbore integ-rity of gas well and bringing a big risk for subsequentproduction operations.1,3–8

In order to reveal the mechanism of casing failure inthe volume fracturing process, this article analyzes thetypical well X201 with casing failure in the Changning-Weiyuan national demonstration zone of shale gas.Through a large number of rock mechanical experi-ments and numerical simulation experiments, the effectof crack angle and crack number on rock mechanicalproperties and its effect on casing failure in the volumefracturing process are studied. Moreover, a compara-tive analysis is conducted with the field multi-arm cali-per imaging tool (MIT) logging curve of casing failuresection in X201 well.

Analysis on the change of rock mechanicalproperties during volume fracturing

Basic data of X201 well

Located in Yibin, Sichuan Province, X201 is in the eastwing of the top Middle Ordovician Shangluochangnose uplift in Changning anticline structure. The targetlayer is a part of Longmaxi Formation in Silurian.

X201 is a vertical shale gas well with measured depth(MD) of 2542m. Completion is conducted withF139.7mm 3 L10.54mm casing with steel grade P110.IBC (Isolation Scanner, post-casing imager) cementingquality evaluation is good without significant channel-ing. According to logging data, elasticity modulus ofthe formation is 13–46GPa, and average Poisson’sratio is 0.23 at 2200–2550m; the minimum horizontalstress is 45–50MPa, and the maximum horizontalstress is 50–70MPa, as shown in Figure 1.

Volume transformations are conducted twice at 2400–2525m in the well. The fracturing fluid is injected directlyinto the formation through the casing, and the two-staged

amount is 1953.5 and 1239.9m3, respectively. The detailedfracturing parameters are shown in Table 1.

Sticking occurs when the gauge tool (F114mm) isrun to make a wiper trip at 2441.6m after two fracturingoperations. Then, casing deformation failure may haveoccurred. MIT logging is subsequently conducted toperform an accurate inspection of the inside of casing,and it is found that serious casing deformation occursnear 2441m. Casing deformation cannot be effectivelyprevented by simply increasing steel grade and wallthickness. It is initially thought that casing deformationfailure may be caused by the degradation of fissuredrock mechanical properties and the redistribution ofnear-well geostress field after the volume fracturing.

Fissured rock theory

Volume fracturing is to form the previous relativelyintact rock in certain volume area into fissured rock with

Table 1. Fracturing operation parameters of X201.

Stage Well section (m) Fracturing fluid (m3) Operation pressure (MPa) Delivery capacity (m3/min) Backflow rate (%)

1 2479–2525 1953.5 70.3 8.5 30.82 2400–2479 1239.9 60.8 10.1 54.5

Figure 1. Interpretation of rock mechanical parameters andin-situ stresses in X201.

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many multiple directional cracks, which changes themacroscopic mechanical properties. According to therelated theories of fracture mechanics and several frac-ture criteria,9,10 rock damage is characterized by the pro-gressive degradation of rock mechanical properties,leading to material failure. The critical stress criterion ofrock damage can be represented as given in equation (1)

sn

smaxn

� �2

+ss

smaxs

� �2

+st

smaxt

� �2

= 1 ð1Þ

Rock damage is assumed to initiate when the maxi-mum nominal stress ratio (as defined in the expressionabove) reaches a value of 1.

A scalar damage variable, d, represents the overalldamage in the material. It initially has a value of 0. Ifrock damage evolution is modeled, d monotonicallyevolves from 0 to 1 upon further loading after the initia-tion of damage. The elasticity modulus and compressivestrength are affected by the damage according to equa-tions (2) and (3).

The linear degradation criterion of elasticity modu-lus is

E=(1� d)3 E0 ð2Þ

The linear degradation criterion of compressivestrength is

sc =(1� d)3 sc0 ð3Þ

The formula to calculate the damage factor d is

d =d f

m 3 (dmaxm � d0

m)

dmaxm 3 (d f

m � d0m)

ð4Þ

From equations (1)–(4), it is shown that rock is con-tinuously damaged due to new cracks formed in

fissured rock during the volume fracturing process,resulting in such mechanical properties’ degradation asthe elasticity modulus, compressive strength, and so on.However, the complexity of fracture networks formedby volume fracturing makes it hard to obtain relativedamage parameters. Hence, there is no quantitativerepresentation model for mechanical properties’ degra-dation. In order to solve this problem, this article car-ried out rock mechanical experiments, aiming at theLongmaxi Formation where casing failure occurred fre-quently. Mesoscopic numerical simulations based onVoronoi diagram were also conducted to analyze thechange law of macroscopic mechanical properties dur-ing volume fracturing process.

Rock mechanical experiment of LongmaxiFormation

In order to reveal the mechanism of casing failure at2400–2525m in the well, the mechanical properties offissured rock in the volume fracturing process werestudied in the first place to establish the numericalinversion model in the fracturing process.

Cores in X201 well in Longmaxi Formation wereselected for the triaxial compression experiment withthe confining pressure of 53MPa in the article. Theexperimental results as follows show the mechanicalproperties of intact rock before fracturing. The elasti-city modulus is 22.6GPa, the yield strength is 161MPa,and the compressive strength is 262MPa. The rocksample appears with an obvious breaking plane with aclear brittle fracture characteristic, as shown inFigure 2(a). The cohesion and the internal frictionangle of rock in Longmaxi Formation are 15MPa and43�, respectively. Such rocks will form into fissuredrocks with multiple cracks under the effect of large-scale hydraulic fracturing.

Figure 2. Test results of compression experiment in Well X201 in Longmaxi Formation: (a) cores comparison and (b) stress–straincurve.

Lin et al. 3

Study on the change of fissured rock mechanicalproperties after volume fracturing

Volume fracturing aims to increase the transformationvolume of shale gas reservoirs to the utmost extent andto form more complex crack networks. It significantlyincreases flow conductivity of reservoirs on one handand divides the intact rock before fracturing into sev-eral small portions on the other. Mechanical properties,including elasticity modulus and strength, varied afterfracturing and the original mechanical condition of cas-ings changed. This article aims to study the effect ofthe formation with complex crack network on rockmechanical properties in the volume fracturing process.Therefore, the model was established with mesoscopicnumerical simulation method based on the Voronoidiagram. The quantitative relationship of the effect ofdifferent crack angles and the number of rock mechani-cal properties was analyzed.3,11–16

Effect of crack angle on the rock mechanicalproperties. Cracks with different angles are formed inthe rock in the volume fracturing process. To obtainthe effect of crack with different angles on rockmechanical properties, the finite element (FE) modelwas established, as shown in Figure 3. The numericalexperiments of the effect of a single crack under differ-ent angles on rock mechanical properties were con-ducted. Wherein areas of different colors are randomlygenerated Voronoi diagrams, representing differentparticle regions in the rock whose elasticity modulus,Poisson’s ratio, cohesion, and internal friction angleare 22645MPa, 0.22, 15MPa, and 43�, respectively (asobtained in the triaxial compression experiment above).The FE model is a square (a=100mm) and the crackis 25mm long. The bottom AB is constrained, and the

confining pressure st is set as 53MPa. Load(pc=300MPa) is imposed on the top edge with linearloading of 100 steps.

According to the FE model shown in Figure 3,numerical simulation experiments for different crackangle u (0�, 15�, 30�, 45�, 60�, 75�, and 90 �) were con-ducted, respectively, with the same method of dataextraction as the triaxial compression experiment. Theengineering stress–stain curves with different crackangles are shown in Figure 4, where the vertical axisrepresents stress (stress= pc) and the horizontal axisrepresents strain (strain=Dd/d).

From Figure 4, it can be known that there are threestages in the failure process of rock mass: elasticitystage, yield stage, and failure stage. Compared to theresults of triaxial compression test in Figure 2, thestress–stain curve of numerical simulations withoutcracks in Figure 4 is the same as that of the test basi-cally. When the single crack in the rock is Lc long, thereis varying change of stress–strain curves in differentdegrees. When u equals 90�, the stress–strain curve sub-stantially coincides with the one without cracks, indi-cating that the crack basically does not influence therock strength. When u is 0�, the stress–strain curve isslightly lower than that without cracks and has asmaller amplitude of variation by contrast. When u is45�, the stress–strain curve drops most drastically com-pared to that without cracks, indicating minimum rockstrength. When u is other value, the stress–stain curvefor a single crack is between that without cracks andthe one when u is 45�.

Taking area I in Figure 4(a) as the yield strength ofthe rock mass, a partial enlarged view of the area isshown in Figure 4(b); taking area II as the compressivestrength of the rock mass, the yield strength and thecompressive strength values are obtained with no cracksand the cracks at different angles. The yield strengthsy0 and the compressive strength sc0 of rock mass with-out cracks are 161.1 and 260.5MPa, respectively. Thesymbols syu and scu are the variables depended on thecrack angle u, representing the yield strength and thecompressive strength at different crack angles. To char-acterize the quantitative change relationship of strengthbetween the fissured rock and the intact rock, the ratiosof sys/sy0 and scs/sc0 are used. Similarly, the elasticitymodulus ratio of Eu to E0 is obtained and analyzed.Eventually, the ratio changing curves of three kinds ofparameters with u are shown in Figure 5.

As can be seen from Figure 5, the curve of elasticitymodulus ratio (Eu/E0) shows a wing-shape variationwhen the crack angle changes from 0� to 90�. With theincrease in the angle, the ratio first decreases and thenincreases. The curve gradually declines from both endsto the middle part and is basically symmetrical aboutthe line (u=45�). The yield strength ratio (syu/sy0) andthe compressive strength ratio (scu/sc0) also show a

Figure 3. Finite element model of crack angle on rockmechanical properties.

4 Advances in Mechanical Engineering

similar trend, and the variation amplitude of compres-sive strength is the most obvious. When the crack angleu is 45�, the ratios of elasticity modulus, yield strength,and compressive strength show a minimum value. Theyare 0.84, 0.83, and 0.72, respectively. This shows a sig-nificant effect of crack angle on the rock mechanicalproperties. Comparison with the previous researchesverifies the reliability and accuracy of the establishedFE model.11,15

Effect of the number of crack on the rock mechanicalproperties. Many cracks are formed in the rock mass inthe volume fracturing process. In order to obtain theeffect of the crack number on the rock mechanicalproperties, the ultimate minimum strength of fissuredrock based on the study of the single crack was studied.The numerical experimental model for the rock masswith multiple cracks at the angle u of 45� was estab-lished, as shown in Figure 6.

The material mechanical parameters, crack attribu-tion, boundary conditions, and loads are the same withthat of the FE model in Figure 3. Numerical simulationexperiments for cracks with different number n (1, 2, 3,4, 5, 6, 7, 8, 9, 16, 32, and 64) were conducted, respec-tively, and stress–stain curves with different crack

numbers are shown in Figure 7. The symbols syn andscn are the variables depended on the crack number n,representing the yield strength and the compressivestrength with different crack numbers. Through thecomparative analysis, the stress–stain curves showvarying degrees of reduction as the crack numberincreases. The stress–stain curve of the rock mass is sig-nificantly lower when n is 2 than that when n is 1, indi-cating a great amplitude vibration. The stress–staincurve keeps a lower level when n is 3; however, theamplitude is smaller than before. When n is more than8, the stress–strain curve substantially coincides, indi-cating basically no change in rock strength. The rockstrength is reduced to the limit. When n is 64, the yieldstrength syn and the compressive strength scn are closeto 115.7 and 176.7MPa, respectively.

As can be seen from Figure 8, with the increasingnumber of cracks, the elasticity modulus ratio En/E0

gradually decreases, with a declining amplitude, to a

Figure 4. Stress–strain curves of rock under different crack dip angles: (a) stress–strain curve and (b) yield strength.

Figure 5. Curve of crack dip angle on rock mechanicalproperties.

Figure 6. Finite element model of crack number on rockmechanical properties.

Lin et al. 5

constant. The ratios of yield strength and compressivestrength also show similar variation trend, of which, thevariation amplitude of the yield strength is most obvi-ous. When n is more than 8, the ratios of elasticity mod-ulus, yield strength, and compressive strength becomestable (0.70, 0.71, and 0.68). This shows that the varia-tion in crack number has no significant effect on therock mechanical properties in the volume fracturingprocess when n reaches a certain value.

To sum up, cracks with different angle and numbercan be formed in the fracturing area in the volume frac-turing process of shale gas and the mechanical strengthof macroscopic rock continues to decrease. Throughnumerous rock mechanical and numerical experiments,the quantitative relationship and the change law ofmechanical properties in X201 well during volume frac-turing are obtained, providing a theoretical basis forthe FE analysis of subsequent casing failure.

Study on the mechanism of casingdeformation failure after fracturing

Establishment of the FE model of casing failure inX201 well

Based on the geological data, logging data, fracturingoperation conditions of X201 well, combined with the

theory of rock damage mechanics and elastic–plastictheory, aiming at the position of casing failure, thethree-dimensional (3D) FE model in X201 well duringvolume fracturing was established (200m long, 5mwide, and 5m thick), as shown in Figure 9.

Six surfaces of the model are the far-field boundaryconditions. Experimental and logging data rockmechanics in Figure 1 are employed to set initialmechanical parameters before volume fracturing.

This article has set up multi-analysis steps to simu-late each stage of the volume fracturing with FE soft-ware. In the process of volume fracturing, the basicparameters of fracturing fluid pressure in casing andcorresponding fracturing areas (pore pressure) areshown in Table 1.

*Field is a kind of keyword to specify predefinedfield variable values, which could be used to adjust thematerial parameters through changing the values ofdependent variables indirectly used in the analysis.Using that keyword and secondary development,degraded material properties are reset to the fracturingareas dynamically in the analysis according to quantita-tive relationship above.

Analysis of the FE numerical simulation

Based on the established FE model of formation-cement sheath-casing in the volume fracturing processof shale gas (see Figure 9), von Mises stress distributioncontours of casing near the damaged position aftertwo-staged fracturing are shown in Figure 10.

As can be seen in Figure 10, the maximum stress ofthe casing reaches 806.3MPa after two-staged fractur-ing and some elements of casing are in the yieldstage; the casing bends obviously at 2440–2442m andelliptical deformation occurs in the casing profile.Qualitatively, the large deformation near 2441m madethe running of rigid gauge tool difficult after the vol-ume fracturing.

Figure 7. Stress–stain curve of rock mass with different crack number: (a) stress–strain curve and (b) yield strength.

Figure 8. Curve of crack number on rock mechanicalproperties.

6 Advances in Mechanical Engineering

To further quantitatively compare and analyze thecasing deformation, the cross section of the casing isdivided into 12 equal parts based on the calculation.The distance between ends (phase angle of 180�) is theinner diameter of deformed casing labeled A–F, respec-tively, as shown in Figure 11. It can simulate the resultthat is similar to that of the field MIT logging curveand compare the accuracy of numerical simulationresults more directly and clearly.

By comparing MIT logging curves, as shown inFigure 11, it can be seen that severe casing deformationdoes occur at 2440–2442m. The minimum inner dia-meter is about 110.8mm, and the maximum inner dia-meter is about 126.1mm. The casing ovality1 can becalculated by equation (5)

z =2(Dmax � Dmin)

(Dmax+Dmin)3 100% ð5Þ

The maximum casing ovality is 12.9%. The resultsaccord with that of MIT logging curve, accuratelysimulating the position and amplitude of casing defor-mation failure. It proves that the model and method inthe article are correct and effective.

Thus, casing failure of X201 well is caused by theelliptical deformation of casing cross section led by vol-ume fracturing. The F114-mm gauge tool cannot passthrough the damaged casing whose inner diameter isonly 110.8mm, and sticking occurred at 2441.6m. Thecasing failure mechanism during volume fracturing ofshale gas wells is revealed.

Figure 9. 3D finite element model during volume fracturing in X201 well.

Figure 10. FE results of damaged casing section.

Lin et al. 7

The study above shows that the casing deformationfailure during volume fracturing is mainly resultedfrom large ovality caused by the extrusion and shearingduring volume fracturing process. Therefore, increasingsteel grade cannot solve the elliptical deformation ofcasing. Instead, increasing the flexural strength byincreasing wall thickness can effectively improve theresistance to ovality. In addition, the reasonable spac-ing design of volume fracturing can also help solve cas-ing deformation failure. The optimal spacing designbased on geological data, rock properties, and in-situstress filed should be applied instead of the simple uni-form one.

Conclusion

In this article, aiming at the effect of complex cracknetworks on rock mechanical properties in the volumefracturing process, the model was established withmesoscopic numerical simulation method based on theVoronoi diagram, and the quantitative relationship ofthe effect of different crack angles and the number ofrock mechanical properties was analyzed.

Through a large number of rock mechanical experi-ments and numerical experiments, this article obtainedthe quantitative relation between the macroscopicmechanical strength of rock and the mesoscopic

characteristics of crack in fissured rock mass. It isfound that rock mechanical properties reduce mostwhen the angle of single crack in Longmaxi Formationis 45�; rock mechanical strength tends to be stable whenthe crack number is more than 8.

The numerical simulation for casing failure duringthe volume fracturing was conducted, and the obtainedmechanism of casing failure is elliptical deformation ofcasing section. Oversize ovality leads to sticking of sub-sequent run tools. Comparison with the field MIT log-ging data verifies the effectiveness of the method usedin this article and the accuracy of calculations.

Increasing wall thickness is more effective to improvethe resistance to ovality than increasing steel grade. Inaddition, the reasonable spacing design of volume frac-turing can also help solve casing deformation failure.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This study was supported by the National Natural

Science Foundation of China (nos 51504207 and 51574198),

Figure 11. Comparison diagram between simulation and MIT logging data.

8 Advances in Mechanical Engineering

Research Fund for the Doctoral Program of HigherEducation of China (no. 20135121110005), and Key Projectof Natural Science of Sichuan Education Department (no.14ZA0037).

References

1. Lian Z, Yu H, Lin T, et al. A study on casing deforma-tion failure during multi-stage hydraulic fracturing forthe stimulated reservoir volume of horizontal shale wells.J Nat Gas Sci Eng 2015; 23: 538–546.

2. Wang H, Liao X and Ding H. Monitoring and evaluat-

ing the volume fracturing effect of horizontal well. J Nat

Gas Sci Eng 2015; 22: 498–502.3. Chen Z, Wang Z and Zeng H. Status quo and prospect

of staged fracturing technique in horizontal wells. Nat

Gas Ind 2007; 27: 78–80.4. Hossain MM, Rahman MK and Rahman SS. Volu-

metric growth and hydraulic conductivity of naturallyfractured reservoirs during hydraulic fracturing: a casestudy using Australian conditions. In: Proceedings of theSPE annual technical conference and exhibition, Dallas,TX, 1–4 October 2000, paper no. SPE 63173. Moskva,Russia: SPE.

5. Tang Y, Wang G, Li Z, et al. Practice and understandingon multiple crack volume fracturing in open hole hori-zontal well section of Zone Su53. Oil Drill Prod Technol

2013; 35: 63–67.6. Yang Q, Liu H, Yan J, et al. Multi-Stage horizontal frac-

turing in casing for low permeability reservoirs. In: Pro-ceedings of the SPE Asia Pacific oil and gas conference

and exhibition, Jakarta, Indonesia, 22–24 October 2013,paper no. SPE 166671. Moskva, Russia: SPE.

7. Yu H, Lian Z and Lin T. Finite element analysis of fail-ure mechanism of casing during shale gas fracturing.China Petrol Mach 2014; 42: 84–88.

8. Brantley SL, Yoxtheimer D, Arjmand S, et al. Waterresource impacts during unconventional shale gas devel-opment: the Pennsylvania experience. Int J Coal Geol

2014; 126: 140–156.9. Camanho P and Davila D. Mixed-mode decohesion finite

elements for the simulation of delamination in composite

materials. Report NASA/TM-2002-211737, June 2002,pp.1–37. Washington, DC: National Aeronautics andSpace Administration.

10. Turon A, Camanho P, Costa J, et al. A damage modelfor the simulation of delamination in advanced compo-sites under variable-mode loading. Mech Mater 2006; 38:1072–1089.

11. Li Y and Yao C. Simulation of rocks containing twocoplanar numerical fissures under uniaxial compressionnon-persistent. Water Resour Power 2014; 32: 127–130.

12. Yao C, Jiang Q, Shao J, et al. A mesoscopic numericalmodel for simulation of rock fracturing. Chin J RockMech Eng 2013; 32(Suppl. 2): 3146–3153.

13. Ghazvinian E, Diederichs M and Quey R. 3D randomVoronoi grain-based models for simulation of brittle rockdamage and fabric-guided micro-fracturing. J Rock Mech

Geotech Eng 2014; 6: 506–521.14. Liu C, Liu H, Zhang Y, et al. Optimal spacing of staged

fracturing in horizontal shale-gas well. J Petrol Sci Eng

2015; 132: 86–93.15. Yao C, Li Y, Jiang Q, et al. Mesoscopic model of failure

process of interlayered rock under compression. Chin JRock Mech Eng 2015; 34: 1–10.

16. Zhang D, Zhang L, Guo J, et al. Research on the pro-duction performance of multistage fractured horizontalwell in shale gas reservoir. J Nat Gas Sci Eng 2015; 26:279–289.

Appendix 1

Notation

d damage variable (dimensionless quantity)Dmax maximum casing external diameter (mm)Dmin minimum casing external diameter (mm)E0 initial elasticity modulus (no damage)

(MPa)E elasticity modulus after rock damage

(MPa)

sc compressive strength after rock damage(MPa)

sc0 initial compressive strength (no damage)(MPa)

sn normal stress of rock material (MPa)ss, st tangential stresses of rock material (MPa)smax

n critical stress of normal direction when therock initiates damage (MPa)

smaxs , smax

t two critical tangential stresses when therock initiates damage (MPa)

dmaxm maximum value of the effective

displacement attained during the loadinghistory (m)

dfm effective displacement at completely

damaged (m)d0m effective displacement at damage

initiation (m)z casing section ellipticity (dimensionless)

Lin et al. 9